Abstract : Multiscale phenomena are ubiquitous in science and engineering, and many multiscale problems are modeled by partial differential equations with general rough coefficients. Direct numerical solution of such problems is often infeasible because a huge number of degrees of freedom are needed to resolve all details on all relevant scales. Numerical multiscale methods aim at reducing the computational cost by efficiently incorporating physically important fine-scale information into a coarse-grid representation. The scope of this minisymposium is to bring together experts in this field to present recent advances in the design and analysis of numerical multiscale methods.
[03763] Super-Localized Generalized Finite Element Method
Format : Talk at Waseda University
Author(s) :
Moritz Hauck (University of Augsburg)
Philip Freese (Technical University Hamburg)
Tim Keil (University of Münster)
Daniel Peterseim (University of Augsburg)
Abstract : We present a multi-scale method for elliptic PDEs with arbitrarily rough coefficients. The method constructs operator-adapted solution spaces with uniform algebraic approximation rates by combining techniques from numerical homogenization and partition of unity methods. Localized basis functions with the same super-exponential localization properties as the Super-Localized Orthogonal Decomposition (SLOD) allow for an efficient implementation of the method. We derive higher-order versions of the method and demonstrate its application to high-contrast channeled coefficients and Helmholtz problems.
[03861] An efficient multiscale approach for simulating Bose-Einstein condensates
Format : Talk at Waseda University
Author(s) :
Christian Döding (Ruhr-University Bochum)
Patrick Henning (Ruhr-University Bochum)
Johan Wärnegård (Columbia University)
Abstract : In this talk we consider the numerical treatment of nonlinear Schrödinger equations as they appear in the modeling of Bose-Einstein condensates. We give numerical examples that demonstrate the influence of the discrete energy on the accuracy of numerical approximations and that a spurious energy can create artificial phenomena such as drifting particles. In order to conserve the exact energy of the equation as accurately as possible, we propose a combination of a class of conservative time integrators with a suitable multiscale finite element discretization in space. This space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture general time invariants with a 6th order accuracy with respect to the chosen mesh size H. This accuracy is preserved due to the conservation properties of the time stepping method. The computational efficiency of the method is demonstrated for a numerical benchmark problem with known exact solution, which is however barely solvable with traditional methods on long time scales.
[04037] Multigrid/multiscale solver for the radiative transfer equation in heterogeneous media
Format : Talk at Waseda University
Author(s) :
QINCHEN SONG (Shanghai Jiao Tong University)
Abstract : The radiative transfer equation describes the interaction between particles and media such as gases, semitransparent liquids , solids, and porous materials, which is widely used in nuclear engineering, thermal radiation transport, etc. In the first part of our work, we construct a multigrid scheme for 1D neutron transport equation based on a second-order discretization scheme that is uniform with respect to $\epsilon$ in the diffusion regime and valid up to the boundary layer and interface layer. We prove its multigrid convergence theoretically and justify it numerically. This multigrid scheme is special in a way that the smoothing procedure in the typical multigrid method can be skipped, which saves a large amount of computation. The 1D scheme can be adapted to the 2D case, and the resulting 2D scheme performs well in the diffusion regime. In the second part, we focus on the 2D radiative transport equation in the transport regime. After discretization of the equation, we get a sparse linear system of extremely high dimensions. Typically, we have 3 ways to solve the linear system: direct methods, iterative methods, and rank-structured methods, which requires computation cost of $I^{6}$, $I^{3}$ and $I^{3}$ respectively (provided that the physical domain is discretized as a $I\times I$ grid). In our work, we use a hybrid scheme of iterative methods and rank-structured methods and reduce the computation cost down to $I^{5/2}$. This is joint work with Min Tang (SJTU) and Lei Zhang (SJTU).
[03533] EXPONENTIALLY CONVERGENT MULTISCALE METHODS FOR HIGH FREQUENCY HETEROGENEOUS HELMHOLTZ EQUATIONS
Format : Talk at Waseda University
Author(s) :
Thomas Y Hou (California Institute of Technology)
Yifan Chen (California Institute of Technology)
Yixuan Wang (California Institute of Technology)
Abstract : We present a multiscale framework for solving the high frequency Helmholtz equation in heterogeneous media without scale separation. Our methods achieve a nearly exponential rate of convergence without suffering from the well-known pollution effect. The key idea is a coarse-fine scale decomposition of the solution space that adapts to the media property and wavenumber. The coarse part is of low complexity while the fine part is local such that it can be computed efficiently.
[03853] A high-order method for elliptic multiscale problems
Format : Talk at Waseda University
Author(s) :
Roland Maier (University of Jena)
Abstract : We present a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The method allows for suitable localization and does not rely on additional assumptions on the domain, the diffusion coefficient, or the exact (weak) solution as typically required for high-order approaches. Rigorous a priori error estimates with respect to the involved discretization parameters are presented and the interplay between these parameters as well as the performance of the method are studied numerically.
[04038] Hierarchical Attention Neural Operator for Multiscale PDEs
Format : Talk at Waseda University
Author(s) :
Bo Xu (Shanghai Jiao Tong University)
Abstract : Complex nonlinear interplays of multiple scales give rise to many interesting physical phenomena and pose significant difficulties for the computer simulation of multiscale PDE models in areas such as reservoir simulation, high-frequency scattering, and turbulence modeling. In this talk, we apply hierarchical attention to a data-driven operator learning problem related to multiscale partial differential equations. An empirical H1 loss function is proposed to counteract the spectral bias of the neural operator approximation for the multiscale solution space. We perform experiments on the multiscale Darcy Flow, Helmholtz equation and Navier-Stokes equation. Our model exhibits noticeably higher accuracy compared to the current neural operator techniques, and it produces state-of-the-art results across a variety of datasets. This is joint work with Xinliang Liu (KAUST) and Lei Zhang (SJTU).
[04167] An abstract framework for multiscale spectral generalized FEMs
Format : Online Talk on Zoom
Author(s) :
chupeng ma (Great Bay University)
Abstract : We present an abstract framework for multiscale spectral generalized FEMs based on locally optimal spectral approximations. A higher convergence rate for the local approximations than previously established is derived under certain conditions. The abstract theory is applied to various problems with strongly heterogeneous coefficients, including convection-diffusion problems, elasticity problems, high-frequency wave problems (Helmholtz, elastic wave, and Maxwell's equations), and fourth-order problems, both in the continuous and discrete settings.
[03857] Multiscale multicontinuum problems in fractured porous media: dimension reduction and decoupling
Format : Talk at Waseda University
Author(s) :
Maria Vasilyeva (Texas A&M University-Corpus Christi)
Abstract : We consider the coupled system of equations that describe flow in fractured porous media. To describe such types of problems, multicontinuum and multiscale approaches are used. The presented decoupling technique separates equations for each continuum that can be solved separately, leading to a more efficient computational algorithm with smaller systems and faster solutions. This approach is based on the additive representation of the operator with semi-implicit approximation by time, where the continuum coupling part is taken from the previous time layer. We extend and investigate this approach for multiscale approximation on the coarse grid using the nonlocal multicontinuum method. We show that the decoupled schemes are stable, accurate, and computationally efficient.
